Nonparametric density estimation is a difficult task in statistics. It is even more difficult for small sample data. For each \(x\)  in the support of a density \(f\)  in a nonparametric
model, the information for this one-dimensional parameter \(f(x)\)  is zero (see [1]}).
[2]} also showed that there is no nonparametric model
for which this information is positive. Properly reducing the infinite dimensional parameter to a finite dimensional one is necessary. To estimate an unknown smooth function as the nonparametric component of a non- and semi-parametric model, as we have done in empirical likelihood we usually approximate it by a step-function and parameterize it using the jump sizes of the step-function.
This approach gives an efficient estimate of the underlying cumulative distribution function. Because this estimate is a step-function, we have to use kernel or other method to smooth it to obtain a density estimate. However, kernel density is actually the convolution of the scaled kernel and the underlying distribution to be estimated. There is always trade-off between the bias and variance. In semiparametric problems, the roughness of the step-function approximation could also affect the finite sample performance of the estimates of the parametric components. Instead of approximating the underlying distribution function by a step-function and then smoothing the discretized estimation, [3]} proposed to use
a Bernstein polynomial approximation and to directly and smoothly estimate the underlying distribution using a maximum approximate Bernstein likelihood method. [3]}'s method parameterizes the underlying distribution by the coefficients of the Bernstein polynomial and differs from other Bernstein polynomial smoothing methods which was initiated by [5]} and use empirical distribution to estimate these coefficients. The maximum approximate Bernstein likelihood method has been successfully applied to grouped, contaminated, multivariate, and interval censored data [6]}, [7]}, [8]}, [9]}. In application to the Cox's proportional hazards regression model, not only a smooth estimate of the survival function but also improved estimates of regression coefficients can be resulted, due to a better approximation of the unknown underlying baseline density function.
